The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: fractals

A geometric or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions.[i]

It was the work of philosopher and mathematician Gottfried Leibniz in the 17th century that first provided groundwork for the early development of fractal geometry.[ii] However, the nature of recursive self-similarity (a core tenet in the study of many fractal-like objects) and the mathematical “monsters”[1] to which this emerging concept gave birth, consequently delayed meaningful research for roughly two centuries.[iii] Nearly two hundred years elapsed before Karl Weierstrass, in 1872, presented the first definition of a function whose graph by today’s standards would be considered fractal. That is, Weierstrass had shown it was possible to define a function with the non-intuitive property that it could be both everywhere continuous and nowhere differentiable.[iv] Subsequent works including those of Georg Cantor (specifically “Cantor sets”), Felix Klein, and Henri Poincaré were crucial in laying a foundation on which much of the modern mathematical investigation of fractal geometry unfolded. Without the aid of modern computational and graphical tools, however, much of the early research in fractal geometry was severely limited. It would be more than a half-century until, in the 1960’s, equipped with the work of his predecessors (i.e., Helge von Koch, Wacław Sierpiński, Pierre Fatou, Gaston Julia, Felix Hausdorff, and Paul Lévy to name a few) did the French-American mathematician Benoit Mandelbrot succeed in uniting hundreds of years of mathematical research by coining the word “fractal”[2] and illustrating his mathematical definition with the aid of remarkable computer-generated visualizations.[v] Most notable among Mandelbrot’s demonstrations was his use of infinite recursion to define what is known today as the Mandelbrot set. Mathematically speaking, the Mandelbrot set is defined to be the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex polynomial zn+1=zn2+c remains bounded. Equivalently, a complex number c is part of the Mandelbrot set if, when starting with z­0=0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.[vi] This seemingly innocuous procedure is responsible for generating the beautiful, infinitely detailed Mandelbrot set pictured below.

Mandelbrot set. Image: Wolfgangbeyer, via Wikimedia Commons.

This figure, and others like it, is generated with the aid of modern computational tools capable of carrying out a myriad of successive iterations far greater in number than the computational limitations of “by-hand” human calculation. The study of infinite descent is also seen in other areas of mathematics such as algebraic geometry (specifically Fermat’s Last Theorem). In these fields, infinite descent is a tool by which the existence of certain special triangles may be determined. For example, infinite descent may be used to prove that no right-angled triangle having integral side lengths will have a square area. The mysterious nature of objects when examined at their infinite limits is, indeed, quite peculiar.

In addition to producing stunning artistic images, fractal geometry has also found diverse applications in fields such as structural engineering, medicine, telecommunications, urban planning, and more.[viii] Iterative methods have been used to create high-strength construction cables by interweaving a series of thinner wires into thicker ones that, in turn, are used in the next iteration to make even larger and stronger cables than before. This simple procedure, usually no more than a few iterations for most industrial purposes, bears a resemblance to a fractal pattern and allows us to make use of the special properties that fractal geometry offers.

As medical knowledge continues to improve, the usefulness of fractals in curing and identifying health concerns becomes increasingly more apparent. Biomimicry, the concept of deriving inspiration for human designs from the natural world, is currently being employed in an attempt to solve the problem of fluid transport by mimicking the fractal patterns of our blood vessels and lungs.viii

Fractal heat exchanger etched in silicon and designed by Deb Pence at Oregon State University, via Fractal Foundation.

According to researchers at Oregon State University, the above figure can be etched into silicon chips, allowing for a cooling fluid (such as liquid nitrogen) to uniformly flow across the surface of the chip, keeping it cool. Researchers say this fractal pattern was derived from human blood vessels and provides a simple low-pressure system to easily cool sensitive computer chips.viii In this case, the relationship between biomimicry and fractals becomes quite clear. The natural question to ask, then, is to what extent may biomimicry assist researchers in addressing biological and natural phenomena? The future is certainly promising.

While classical Euclidian geometry and other related areas of mathematics are often used to understand and predict natural phenomena, these traditional modes of thinking may prove insufficient in answering some of the more complex questions that arise in nature (some of which have been discussed here). After all, as Benoit Mandelbrot himself once said, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”[ix] It is the heart of fractal geometry that attempts to analyze and understand many of these more complex phenomena.

[1] These “monsters” were mathematical problems of immense complexity that, according to Leibniz and others, were believed to be beyond the scope of contemporary geometric knowledge.

In class, we talked about how dimensions can be non-integer values. We were given some examples of fractals, shapes with non-integer dimensions, and we were able to calculate the dimensions of some fairly simple fractals. But what about more complex shapes, that can’t be easily doubled, quadrupled, or so on? How does one measure, arbitrarily, the dimension of any shape?

Well, one of the ways to do so is to find the Hausdorff Dimension of a set. This concept of measuring dimensions was developed by (big surprise) Felix Hausdorff back in 1918.

The key idea is this: a “circle” with dimension 1 (a line) has its length vary proportionally to its “radius”. A circle with dimension 2 has its area vary proportionally to its radius squared, and so on with spheres and volume. To extrapolate, a “circle” with any dimension p would have its “p-volume” vary proportionally to its radius to the power of p. So, if p were 2.5, a 2.5 dimensional circle would, when doubled, increase in its 2.5 dimensional volume by 22.5 = about 5.66.

Now, hold onto your hats (if you don’t have a hat, go get one real quick, then hold onto it) because here’s where things get interesting. Let’s say you wanted to cover a 3-dimensional object with a bunch of smaller spheres, and measure the “4 dimensional volume” of the spheres. Well, we know that the 4-dimensional volume of an object is proportional to its radius4, so we can get an idea of how the 4-dimensional volume changes by simply adding together the radii4 of every sphere covering the object.

Now let’s take an arbitrary covering of our object by spheres and measure its 4-dimensional volume as above. We will get some number A. Now, how does that number change as our spheres shrink? Let’s say we, for example, replaced each sphere with 8 spheres, each having half the radius, and still managed to cover the object. Note that while the 3-dimensional volume of our spheres has remained the same (8 spheres of radius 1/2 have the same volume as 1 of radius 1), our 4-dimensional volume has been cut in half!

The main idea is that by shrinking the radii of our spheres, we can arbitrary decrease the 4-dimensional volume of the spheres covering our object. Since we can cover our object with spheres having arbitrary small 4-dimensional volume, it would make sense that the object would have 0 4-dimensional volume, which is consistent with how one would think about 4-dimensional area.

The key here is that this is true for any d-dimensional area if d is greater than 3, because our object is 3-dimensional. Thus, if we took the infimum of values for d such that this is true, we would get 3.

To reiterate, we had a 3-dimensional object. We were able to determine it was 3-dimensional because for any dimension d higher than 3, we could cover our object with spheres such that their d-dimensional volume was arbitrarily small. To be very precise:

The Hausdorff content of dimension d of an object is the infimum of numbers δ ≥ 0 such that there is some cover of the object by balls of radius r1, r2,… such that (r1d+r2d+r3d+…)< δ.

The Hausdorfff dimension of an object is the infimum of numbers d such that the hausdorff content of dimension d of an object is equal to 0.

Now, this approach matches quite nicely with our definition of integer dimensions, and provides a very nice way for us to expand that notion into other, non-integer dimensions. For example, this approach can actually be used in a surprising way: to find the dimension of coastlines.

Perhaps you’ve heard of the coastline paradox: that the more precise you try to measure the coastline, the longer it gets, seemingly without bound. This should signal to us that the coastline behaves as a fractal, and since the above method gives us a way to measure the dimension of arbitrary objects, we can use it to try and measure the dimension of the coastline.

Measuring the dimension of the coast of Great Britain. Image: Prokofiev, via Wikimedia Commons.

Here’s how it’s done: first, cover the coastline in large circles and measure the sum of the radii all put to some power p. Then, shrink the circles and measure again the sum of the radii all put to the power p. If this number keeps getting smaller, you’ve overestimated the dimension of the coast. If the number keeps getting bigger, you’ve underestimated it. If you keep doing this, you can fine-tune your estimate of the dimension. In fact, Mandelbrot did this very thing, and got that the coast of Great Britain had a fractional dimension of about 1.25, while the coastline of South Africa had a fractional dimension of about 1.02. While these are just estimates, it’s still cool to see how abstract ideas such as this can be used to measure things in real life.

We’ve all heard of the Butterfly Effect or maybe seen the movie called “Chaos Theory.” Unfortunately, Hollywood has led us all astray once again. The Butterfly Effect is a small part of Chaos Theory, but that’s not all it is. Chaos Theory is simply a mathematical process that helps us understand large and complex data models. Due to the sheer size of the mathematical models involved, a computer is required to simulate them. This is why Chaos Theory was not “invented” until the late 20th century; we needed to invent the computer first! Examples requiring computer assisted Chaos Theory simulations include weather forecasting, migratory patterns of birds and many other areas.

The Beginning

It all started in the year 1960, when a man named Edward Lorenz built a weather model on his computer at the Massachusetts Institute of Technology. Given an initial set of parameters, it would compute a series of weather conditions that would never repeat. This was revolutionary in his time, and some people thought that by providing the current weather parameters, Lorenz’s program could predict the weather perfectly. Amidst all of the commotion Lorenz decided to “restart” a simulation he had run previously, but he didn’t start from the beginning as he had before, he started from a point near the middle of the simulation. The reason this was an issue was because the computer he was working on had six places of precision when working with numbers, but it only printed out three decimal places. Lorenz then used the three decimal numbers and used them as his initial starting conditions. Given the small differences between the numbers (going from six digit precision to three) he obtained vastly different results. These results were very surprising to Lorenz, but he eventually figured out his mistake. By that time, Chaos Theory was born.

Chaos Theory Subjects

There are a few different principles for the Chaos theory that are quite interesting. One area that I’m sure the majority of people have heard of is fractals. Fractals are essentially a pattern that repeats forever. These images can be very, very complex, like the one below.

Image: Public domain, via Wikimedia Commons.

Fractals relate to Chaos Theory because they are greatly influenced by the initial conditions. Supply a parameter that is slightly different from a previous image will result in a drastically different pattern. Another interesting thing about fractals is they appear in nature as well. If you’ve ever looked at an image where a snowflake structure has been magnified, it’s easy to see that there is a repeating pattern in the formation. Natural fractal formations also appear in the formation of tree branches and leaves, as well as some types crustacean shells.

Another principle of chaos is mixing. Mixing is the process by which two substances (molecules, balloons, anything) that start at the same location, will end up at drastically different end locations. An example of this is a water molecule in a lake. At some point in time there will be two water molecules that start near each other, but in 50 years are on opposite sides of the lake. This type of analogy can be applied to many different applications such as liquids mixing and any simulation that uses a fluid dynamics module (often called a Computational Fluid Dynamics or “CFD”).

Conclusion

Chaos is everywhere around us all the time, whether we like it or not. It crops up in nature and science, where it has interesting applications that cover a wide variety of topics. There is a lot of discussion on whether humankind will ever be able to fully comprehend Chaos Theory, and no one knows the answer. One thing is certain: without overcoming Chaos Theory, we will never be able to predict the weather, which is upsetting to say the least. I would like to end with a quote from Albert Einstein, who said the following:

“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” –Albert Einstein

“Without mathematics there is no art,” said Luca Pacioli, a contemporary of Da Vinci.

Working through the galley division algorithm made me think about the connection between art and math throughout history. I was curious about the earliest connections – how early did people first begin to see beauty in geometric figures and the patterns found in nature? The Golden Ratio, often mentioned as the earliest convergence between art and math, was considered “aesthetically pleasing”. The pyramids and Parthenon rely on special ratios and math in their construction. Da Vinci famously studied the proportionality in the human body and relied on mathematics throughout his artwork, even using his art to solve a mathematical problem that had burdened the great thinkers of his day. Fractals were a fascinating discovery of infinite repetition that both created beautiful figures and contributed a great deal to modern computing and accurate global coast mapping.

Galley, or “Scratch”, Division

This is an instance of beautifully simple algorithm that solves a mathematical problem with two results: the quotient and remainder from a long division problem, and a cleverly drawn shape that resembles a boat. This method of long division was actually the most widely preferred for solving long division for seventeen centuries, popular until around 1600 AD. The process (not to its entire solution) is displayed in the image below, alongside a more artful example from “The Science-History of the Universe” (Francis Rolt-Wheeler 1910). The end result is the quotient displayed to the right of the original dividend, with the remainder shown as the cascading digits above the dividend (those without a dash through them in the modern example)

Galley division by a 16th century Venetian monk, published in the 1910 book The Science-History of the Universe by Francis Rolt-Wheeler. Image: Public domain, via Wikimedia Commons.

The Pyramids and the Parthenon – Egyptian triangle and golden ratio found in its base, Parthenon and golden rectangle found above its pillars

While there isn’t evidence of the Egyptians intentionally using Phi or Pi when designing their structures, both numbers are seen in the pyramids’ dimensions with nearly perfect accuracy. Mock pyramids based on both Phi (the golden ratio) and pi, vary from the actual dimensions of the pyramids by very little, on the order of inches. In terms of Phi, this comes down to 1.4 inches, which can be attributed to rounding differences or differences in the estimated dimensions of the pyramids with their covering stones (the pyramids now are missing their smooth outer shells, so what we see today is the structural core). In terms of Pi, the difference between a pyramid based on Pi and the estimated dimensions of the completed pyramids differ by 5.5 inches. So while the methods by which the Egyptians arrived at the dimensions they used, given that the pyramids were built to within 1/15 of a degree of true north, it is doubtful that the presence of Phi and Pi are pure coincidence.

As with the pyramids, there is doubt about whether there was conscious use of the golden ratio in the construction of the Parthenon in Greece. The Parthenon was finished in 438 BC, and the first documented awareness by the Greeks of the golden ratio was not until 300 BC in Euclid’s Elements. The golden rectangle can be found in its construction, however there is debate that this may be more coincidental or a case of modern mathematical knowledge informing interpretation. When the spiral of the golden ratio is laid over a photo taken of the front face of the Parthenon without perspective distortion, one needs to assume that the bottom of the spiral will rest on the second (not the bottom) step of the structure, however the spiral is perfectly represented in the area directly over the columns.

da Vinci the Vitruvian man

Da Vinci used art to discover a close approximate solution to one of the greatest mathematical problems of his day, squaring the circle. Squaring the circle is the problem of generating a circle and a square with the same area, given only a compass and straightedge to draw both. He did so through the Vitruvian Man, with 99.8% accuracy (further accuracy even today isn’t possible because of the irrational nature of Pi), and created one of the most iconic images of the Renaissance at the same time. The two shapes, the square and the circle, are drawn simply by changing the positions of the man’s arms and legs, generating each shape with a (nearly) equal area to the other.

Credit is due to Marcus Vetruvius , a Roman architect of the first century BC, who recognized two interesting facts: he used the navel as the center point to draw the circle using a compass, and recognized that a man’s arm span and his height almost perfectly coincide, drawing the square. Leonardo was responsible for bringing together these two ideas to solve the problem of squaring the circle. This idea interestingly came at the same time as a resurgence in the philosophy called NeoPlatonism, which places beings in a hierarchy with God at the top, the devil and animals at the bottom, and man directly in the middle, just as the Vetruvian Man formed the center of solving a seemingly irreconcilable problem in mathematics

Fractals

As a computer science major, fractals are one of the most interesting examples of when math and art have crossed paths. The history of fractals begins with a name familiar to anyone who’s been through Calculus – Gottfried Liebniz – who considered the concept of recursive self-similarity. His musings languished as just that for centuries, partly because mathematicians of his time were resistant to unfamiliar, uncomfortable concepts, going so far as to call them “mathematical monsters”. Progress was made in the early 20th century by Helge von Koch, who drew the familiar Koch Snowflake, a simple and yet incredibly complex definition for self-similar geometric figures. The term we use today was not coined until the 1960s, when Benoit Mandelbrot, working at IBM where he had access to some of the earliest computers, researched objects typically considered “messy” and hard to define – things like clouds and shorelines that were considered utterly random. His research proved they could be reduced to self-similar patterns, and this led to more accurate coastline mapping and measurements. The art of fractals began to take shape in the 1980s with the visual representation of the calculations of fractals, using the increasing computing power of modern hardware and fractal calculating software. Fractal images, Julia and Mandelbrot Sets being two of the most recognized, are fascinatingly intricate, crystallizing years of research and theory in beautiful, “aesthetically pleasing” imagery.